Theorem csbiebt 3923

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3927.) (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbiebt	⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbiebt

Step	Hyp	Ref	Expression
1		elex 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		spsbc 3790	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
3	2	adantr 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → [𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶)))
4		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → 𝐴 ∈ V)
5		biimt 360	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ (𝑥 = 𝐴 → 𝐵 = 𝐶)))
6		csbeq1a 3907	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
7	6	eqeq1d 2733	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
8	5, 7	bitr3d 281	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
9	8	adantl 481	. . . . 5 ⊢ (((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) ∧ 𝑥 = 𝐴) → ((𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
10		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ∈ V
11		nfnfc1 2905	. . . . . 6 ⊢ Ⅎ𝑥Ⅎ𝑥𝐶
12	10, 11	nfan 1901	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ Ⅎ𝑥𝐶)
13		nfcsb1v 3918	. . . . . . 7 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵
14	13	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
15		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥𝐶)
16	14, 15	nfeqd 2912	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
17	4, 9, 12, 16	sbciedf 3821	. . . 4 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → ([𝐴 / 𝑥](𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
18	3, 17	sylibd 238	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
19	13	a1i 11	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵)
20		id 22	. . . . . . . 8 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥𝐶)
21	19, 20	nfeqd 2912	. . . . . . 7 ⊢ (Ⅎ𝑥𝐶 → Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
22	11, 21	nfan1 2192	. . . . . 6 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
23	7	biimprcd 249	. . . . . . 7 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → (𝑥 = 𝐴 → 𝐵 = 𝐶))
24	23	adantl 481	. . . . . 6 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → (𝑥 = 𝐴 → 𝐵 = 𝐶))
25	22, 24	alrimi 2205	. . . . 5 ⊢ ((Ⅎ𝑥𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶))
26	25	ex 412	. . . 4 ⊢ (Ⅎ𝑥𝐶 → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
27	26	adantl 481	. . 3 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (⦋𝐴 / 𝑥⦌𝐵 = 𝐶 → ∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶)))
28	18, 27	impbid 211	. 2 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))
29	1, 28	sylan 579	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝐶) → (∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2105  Ⅎwnfc 2882  Vcvv 3473  [wsbc 3777  ⦋csb 3893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475  df-sbc 3778  df-csb 3894

This theorem is referenced by:  csbiedf  3924  csbieb  3925  csbiegf  3927